Abstract
Weakly pumped systems with approximate conservation laws can be efficiently described by (generalized) Gibbs ensembles if the steady state of the system is unique. However, such a description can fail if there are multiple steady state solutions, for example, a bistability. In this case domains and domain walls may form. In one-dimensional (1D) systems any type of noise (thermal or non-thermal) will in general lead to a proliferation of such domains. We study this physics in a 1D spin chain with two approximate conservation laws, energy and the zz-component of the total magnetization. A bistability in the magnetization is induced by the coupling to suitably chosen Lindblad operators. We analyze the theory for a weak coupling strength \epsilonϵ to the non-equilibrium bath. In this limit, we argue that one can use hydrodynamic approximations which describe the system locally in terms of space- and time-dependent Lagrange parameters. Here noise terms enforce the creation of domains, where the typical width of a domain wall goes as \sim 1/\sqrt{\epsilon}∼1/ϵ while the density of domain walls is exponentially small in 1/\sqrt{\epsilon}1/ϵ. This is shown by numerical simulations of a simplified hydrodynamic equation in the presence of noise.
Highlights
In the thermodynamic limit the steady state of an interacting many-body quantum system can be described in a very compact way by a Gibbs ensemble, ρ ∼ e−, N i=1 λi Q i where the Qi areN conserved quantities of the system
In one-dimensional integrable many-particle systems N grows linearly with system size, in this case the term ‘generalized Gibbs ensemble’ is used [1,2,3]. This approach can even be used if the conservation laws are only approximately valid and if the system is weakly driven out of equilibrium as long as scattering processes which conserve the Qi dominate the dynamics
We will investigate a simple 1D model which allows one to study the role of approximate conservation laws, the validity of Gibbs ensembles and the relations of bistabilities and noise in a controlled way
Summary
In the thermodynamic limit the steady state of an interacting many-body quantum system can be described in a very compact way by a Gibbs ensemble, ρ ∼ e−. In one-dimensional integrable many-particle systems N grows linearly with system size, in this case the term ‘generalized Gibbs ensemble’ is used [1,2,3] This approach can even be used if the conservation laws are only approximately valid and if the system is weakly driven out of equilibrium as long as scattering processes which conserve the Qi dominate the dynamics. Starting from a 1D system with just two exact conservation laws (energy and magnetization), we add weak perturbations of strength ε which break the corresponding symmetries and drive the system out of equilibrium. We will investigate a simple 1D model which allows one to study the role of approximate conservation laws, the validity of Gibbs ensembles and the relations of bistabilities and noise in a controlled way. In the following we will only consider the situation where such dark states do not exist, i.e., we consider the case γ < 1 only
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